Dense kernel matrices represented in H 2 matrix format typically require less storage and have faster matrix-vector multiplications than when these matrices are represented in the standard dense format. In this article, we present H2Pack, a high-performance, shared-memory library for constructing and operating with H 2 matrix representations for kernel matrices defined by non-oscillatory, translationally invariant kernel functions. Using a hybrid analytic-algebraic compression method called the proxy point method, H2Pack can efficiently construct an H 2 matrix representation with linear computational complexity. Storage and matrix-vector multiplication also have linear complexity. H2Pack also introduces the concept of “partially admissible blocks” for H 2 matrices to make H 2 matrix-vector multiplication mathematically identical to the fast multipole method (FMM) if analytic expansions are used. We optimize H2Pack from both the algorithm and software perspectives. Compared to existing FMM libraries, H2Pack generally has much faster H 2 matrix-vector multiplications, since the proxy point method is more effective at producing block low-rank approximations than the analytic methods used in FMM. As a tradeoff, H 2 matrix construction in H2Pack is typically more expensive than the setup cost in FMM libraries. Thus, H2Pack is ideal for applications that need a large number of matrix-vector multiplications for a given configuration of data points.

中文翻译：

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H2Pack

表示的密集核矩阵H 2与以标准密集格式表示这些矩阵时相比，矩阵格式通常需要更少的存储空间并且具有更快的矩阵向量乘法。在本文中，我们介绍了 H2Pack，这是一个高性能的共享内存库，用于构建和操作H 2由非振荡、平移不变的核函数定义的核矩阵的矩阵表示。使用称为代理点方法的混合解析代数压缩方法，H2Pack 可以有效地构造一个H 2具有线性计算复杂度的矩阵表示。存储和矩阵向量乘法也具有线性复杂度。H2Pack 还引入了“部分允许块”的概念，用于H 2要制作的矩阵H 2如果使用解析展开，矩阵向量乘法在数学上与快速多极法 (FMM) 相同。我们从算法和软件角度优化 H2Pack。与现有的 FMM 库相比，H2Pack 通常具有更快的速度H 2矩阵向量乘法，因为代理点方法在生成块低秩近似方面比 FMM 中使用的分析方法更有效。作为权衡，H 2H2Pack 中的矩阵构造通常比 FMM 库中的设置成本更昂贵。因此，H2Pack 非常适合需要为给定数据点配置进行大量矩阵向量乘法的应用程序。